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Abstract and Applied Analysis
Volumeย 2012ย (2012), Article IDย 797398, 16 pages
http://dx.doi.org/10.1155/2012/797398
Research Article

Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives

1School of Information and Engineering, Wenzhou Medical College, Zhejiang, Wenzhou 325035, China
2School of Mathematical and Informational Sciences, Yantai University, Shandong, Yantai 264005, China
3Information Engineering Department, Anhui Xinhua University, Anhui, Hefei 230031, China

Received 24 April 2012; Accepted 29 May 2012

Academic Editor: Yonghongย Wu

Copyright ยฉ 2012 Tunhua Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the singular fractional-order boundary-value problem with a sign-changing nonlinear term โˆ’๐’Ÿ๐›ผ๐‘ฅ(๐‘ก)=๐‘(๐‘ก)๐‘“(๐‘ก,๐‘ฅ(๐‘ก),๐’Ÿ๐œ‡1๐‘ฅ(๐‘ก),๐’Ÿ๐œ‡2๐‘ฅ(๐‘ก),โ€ฆ,๐’Ÿ๐œ‡๐‘›โˆ’1๐‘ฅ(๐‘ก)),0<๐‘ก<1,๐’Ÿ๐œ‡๐‘–๐‘ฅ(0)=0,1โ‰ค๐‘–โ‰ค๐‘›โˆ’1,๐’Ÿ๐œ‡๐‘›โˆ’1+1๐‘ฅ(0)=0, ๐’Ÿ๐œ‡๐‘›โˆ’1โˆ‘๐‘ฅ(1)=๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐’Ÿ๐œ‡๐‘›โˆ’1๐‘ฅ(๐œ‰๐‘—), where ๐‘›โˆ’1<๐›ผโ‰ค๐‘›, ๐‘›โˆˆโ„• and ๐‘›โ‰ฅ3 with 0<๐œ‡1<๐œ‡2<โ‹ฏ<๐œ‡๐‘›โˆ’2<๐œ‡๐‘›โˆ’1 and ๐‘›โˆ’3<๐œ‡๐‘›โˆ’1<๐›ผโˆ’2, ๐‘Ž๐‘—โˆˆโ„,0<๐œ‰1<๐œ‰2<โ‹ฏ<๐œ‰๐‘โˆ’2<1 satisfying โˆ‘0<๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1<1, ๐’Ÿ๐›ผ is the standard Riemann-Liouville derivative, ๐‘“โˆถ[0,1]ร—โ„๐‘›โ†’โ„ is a sign-changing continuous function and may be unbounded from below with respect to ๐‘ฅ๐‘–, and ๐‘โˆถ(0,1)โ†’[0,โˆž) is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field.

1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines, and particularly in the mathematical modeling of systems and processes in physics, chemistry, aerodynamics, electrodynamics of complex medium, and polymer rheology [1โ€“6]. Fractional-order models have proved to be more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Hence fractional differential equations have attracted great research interest in recent years, and for more details we refer the reader to [7โ€“16] and the references cited therein.

In this paper, we consider the existence of nontrivial solutions for the following singular fractional-order boundary-value problem with a sign-changing nonlinear term and fractional derivatives: โˆ’๐’Ÿ๐›ผ๐‘ฅ(๐‘ก)=๐‘(๐‘ก)๐‘“(๐‘ก,๐‘ฅ(๐‘ก),๐’Ÿ๐œ‡1๐‘ฅ(๐‘ก),๐’Ÿ๐œ‡2๐‘ฅ(๐‘ก),โ€ฆ,๐’Ÿ๐œ‡๐‘›โˆ’1๐’Ÿ๐‘ฅ(๐‘ก)),0<๐‘ก<1,๐œ‡๐‘–๐‘ฅ(0)=0,1โ‰ค๐‘–โ‰ค๐‘›โˆ’1,๐’Ÿ๐œ‡๐‘›โˆ’1+1๐‘ฅ(0)=0,๐’Ÿ๐œ‡๐‘›โˆ’1๐‘ฅ(1)=๐‘โˆ’2๎“๐‘—=1๐‘Ž๐‘—๐’Ÿ๐œ‡๐‘›โˆ’1๐‘ฅ๎€ท๐œ‰๐‘—๎€ธ,(1.1) where ๐‘›โˆ’1<๐›ผโ‰ค๐‘›,๐‘›โˆˆโ„• and ๐‘›โ‰ฅ3 with 0<๐œ‡1<๐œ‡2<โ‹ฏ<๐œ‡๐‘›โˆ’2<๐œ‡๐‘›โˆ’1 and ๐‘›โˆ’3<๐œ‡๐‘›โˆ’1<๐›ผโˆ’2, ๐‘Ž๐‘—โˆˆโ„,0<๐œ‰1<๐œ‰2<โ‹ฏ<๐œ‰๐‘โˆ’2<1 satisfying โˆ‘0<๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1<1, ๐’Ÿ๐›ผ is the standard Riemann-Liouville derivative, ๐‘“โˆถ[0,1]ร—โ„๐‘›โ†’โ„ is a sign-changing continuous function and may be unbounded from below with respect to ๐‘ฅ๐‘–, and ๐‘โˆถ(0,1)โ†’[0,โˆž) is continuous.

In this paper, we assume that ๐‘“โˆถ[0,1]ร—โ„๐‘›โ†’โ„, which implies that the problem (1.1) is changing sign (or semipositone particularly). Differential equations with changing-sign arguments are found to be important mathematical tools for the better understanding of several real-world problems in physics, chemistry, mechanics, engineering, and economics [17โ€“19]. In general, the cone theory is difficult to handle this type of problems since the operator generated by ๐‘“ is not a cone mapping. So to find a new method to solve changing-sign problems is an interesting, important, and difficult work. An effective approach to this problem was recently suggested by Sun [20] based on the topological degree of a completely continuous field. Then, Han and Wu [21, 22] obtained a new Leray-Schauder degree theorem by improving the results of Sun [20]. In [22], Han et al. also investigated a kind of singular two-point boundary-value problems with sign-changing nonlinear terms by applying the new Leray-Schauder degree theorem obtained in [22].

To our knowledge, very few results have been established when ๐‘“ is changing sign [20โ€“24]. In [20, 21, 23], ๐‘“ permits sign changing but required to be bounded from below. In [22, Theorem 1.1], ๐‘“ may be a sign-changing and unbounded function, but the Green function must be symmetric and ๐‘“ is controlled by a special function โ„Ž(๐‘ข)=โˆ’๐‘โˆ’๐‘|๐‘ข|๐œ‡, where ๐‘>0,๐‘>0 and ๐œ‡โˆˆ(0,1). Recently, by improving and generalizing the main results of Sun [20] and Han et al. [21, 22], Liu et al. [24] established a generalized Leray-Schauder degree theorem of a completely continuous field for solving ๐‘š-point boundary-value problems for singular second-order differential equations.

Motivated by [20โ€“24], we established some new results on the existence of nontrivial solutions for the problem (1.1) by computing the topological degree of a completely continuous field. The conditions used in the present paper are weaker than the conditions given in previous works [20โ€“24], and particularly we drop the assumption of even function in [24]. The new features of this paper mainly include the following aspects. Firstly, the nonlinear term ๐‘“(๐‘ก,๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›) in the BVP (1.1) is allowed to be sign changing and unbounded from below with respect to ๐‘ฅ๐‘–. Secondly, the nonlinear term ๐‘“ involves fractional derivatives of unknown functions. Thirdly, the boundary conditions involve fractional derivatives of unknown functions which is a more general case, and include the two-point, three-point, multipoint, and some nonlocal problems as special cases of (1.1).

2. Preliminaries and Lemmas

In this section, we give some preliminaries and lemmas.

Definition 2.1. Let ๐ธ be a real Banach space. A nonempty closed convex set ๐‘ƒโŠ‚๐ธ is called a cone of ๐ธ if it satisfies the following two conditions:
(1) ๐‘ฅโˆˆ๐‘ƒ,๐œŽ>0 implies ๐œŽ๐‘ฅโˆˆ๐‘ƒ;
(2) ๐‘ฅโˆˆ๐‘ƒ,โˆ’๐‘ฅโˆˆ๐‘ƒ implies ๐‘ฅ=๐œƒ.

Definition 2.2. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.
Let ๐ธ be a real Banach space, ๐ธโˆ— the dual space of ๐ธ, ๐‘ƒ a total cone in ๐ธ, that is, ๐ธ=๐‘ƒโˆ’๐‘ƒ, and ๐‘ƒโˆ— the dual cone of ๐‘ƒ.

Lemma 2.3 (Deimling [25]). Let ๐ฟโˆถ๐ธโ†’๐ธ be a continuous linear operator, ๐‘ƒ a total cone, and ๐ฟ(๐‘ƒ)โŠ‚๐‘ƒ. If there exist ๐œ“โˆˆ๐ธโงต(โˆ’๐‘ƒ) and a positive constant ๐‘ such that ๐‘๐ฟ(๐œ“)โ‰ฅ๐œ“, then the spectral radius ๐‘Ÿ(๐ฟ)โ‰ 0 and has a positive eigenfunction corresponding to its first eigenvalue ๐œ†=๐‘Ÿ(๐ฟ)โˆ’1.

Lemma 2.4 (see [25]). Let ๐‘ƒ be a cone of the real Banach space ๐ธ, and ฮฉ a bounded open subsets of ๐ธ. Suppose that ๐‘‡โˆถฮฉโˆฉ๐‘ƒโ†’๐‘ƒ is a completely continuous operator. If there exists ๐‘ฅ0โˆˆ๐‘ƒโงต{๐œƒ} such that ๐‘ฅโˆ’๐‘‡๐‘ฅโ‰ ๐œ‡๐‘ฅ0,โˆ€๐‘ฅโˆˆ๐œ•ฮฉโˆฉ๐‘ƒ,๐œ‡โ‰ฅ0, then the fixed-point index ๐‘–(๐‘‡,ฮฉโˆฉ๐‘ƒ,๐‘ƒ)=0.
Let ๐ฟโˆถ๐ธโ†’๐ธ be a completely continuous linear positive operator with the spectral radius ๐‘Ÿ1โ‰ 0. On account of Lemma 2.3, there exist ๐œ‘1โˆˆ๐‘ƒโงต{๐œƒ} and ๐‘”1โˆˆ๐‘ƒโˆ—โงต{๐œƒ} such that ๐ฟ๐œ‘1=๐‘Ÿ1๐œ‘1,๐‘Ÿ1๐ฟโˆ—๐‘”1=๐‘”1,(2.1) where ๐ฟโˆ— is the dual operator of ๐ฟ. Choose a number ๐›ฟ>0 and let ๐‘ƒ๎€ท๐‘”1๎€ธ=๎€ฝ,๐›ฟ๐‘ขโˆˆ๐‘ƒโˆถ๐‘”1||||๎€พ(๐‘ข)โ‰ฅ๐›ฟ|๐‘ข|,(2.2) then ๐‘ƒ(๐‘”1,๐›ฟ) is a cone in ๐ธ.

Lemma 2.5 (see [24]). Suppose that the following conditions are satisfied.โ€‰(A1)โ€‰๐‘‡โˆถ๐ธโ†’๐‘ƒ is a continuous operator satisfying limโ€–๐‘ขโ€–โ†’+โˆžโ€–๐‘‡๐‘ขโ€–โ€–๐‘ขโ€–=0;(2.3)โ€‰(A2)โ€‰๐นโˆถ๐ธโ†’๐ธ is a bounded continuous operator and there exists ๐‘ข0โˆˆ๐ธ such that ๐น๐‘ข+๐‘ข0+๐‘‡๐‘ขโˆˆ๐‘ƒ, for all ๐‘ขโˆˆ๐ธ;โ€‰(A3)โ€‰๐‘Ÿ1>0 and there exist ๐‘ฃ0โˆˆ๐ธ and ๐œ‚>0 such that ๐ฟ๐น๐‘ขโ‰ฅ๐‘Ÿ1(1+๐œ‚)๐ฟ๐‘ขโˆ’๐ฟ๐‘‡๐‘ขโˆ’๐‘ฃ0,โˆ€๐‘ขโˆˆ๐ธ.(2.4)Let ๐ด=๐ฟ๐น, then there exists ๐‘…>0 such that ๎€ทdeg๐ผโˆ’๐ด,๐ต๐‘…๎€ธ,๐œƒ=0,(2.5) where ๐ต๐‘…={๐‘ขโˆˆ๐ธโˆถโ€–๐‘ขโ€–<๐‘…} is the open ball of radius ๐‘… in ๐ธ.

Remark 2.6. If the operator ๐‘‡ which satisfies the conditions of Lemma 2.5 is a null operator, then Lemma 2.5 turns into Theoremโ€‰โ€‰1 in [20]. On the other hand, if the operator ๐‘‡ in Lemma 2.5 is such that there exist constants ๐›ผโˆˆ(0,1) and ๐‘>0 satisfying ||๐‘‡๐‘ข||โ‰ค๐‘||๐‘ข||๐›ผ for all ๐‘ขโˆˆ๐ธ, then Lemma 2.5 turns into Theoremโ€‰โ€‰2.1 in [22] or Theoremโ€‰โ€‰1 in [21]. So Lemma 2.5 is an improvement of the results of paper [20โ€“22].
Now we present the necessary definitions from fractional calculus theory. These definitions can be found in some recent literatures, for example, [26, 27].

Definition 2.7 (see [26, 27]). The Riemann-Liouville fractional integral of order ๐›ผ>0 of a function ๐‘ฅโˆถ(0,+โˆž)โ†’โ„ is given by ๐ผ๐›ผ1๐‘ฅ(๐‘ก)=๎€œฮ“(๐›ผ)๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1๐‘ฅ(๐‘ )๐‘‘๐‘ (2.6) provided that the right-hand side is pointwisely defined on (0,+โˆž).

Definition 2.8 (see [26, 27]). The Riemann-Liouville fractional derivative of order ๐›ผ>0 of a function ๐‘ฅโˆถ(0,+โˆž)โ†’โ„ is given by ๐’Ÿ๐›ผ1๐‘ฅ(๐‘ก)=๎‚€๐‘‘ฮ“(๐‘›โˆ’๐›ผ)๎‚๐‘‘๐‘ก๐‘›๎€œ๐‘ก0(๐‘กโˆ’๐‘ )๐‘›โˆ’๐›ผโˆ’1๐‘ฅ(๐‘ )๐‘‘๐‘ ,(2.7) where ๐‘›=[๐›ผ]+1, and [๐›ผ] denotes the integer part of the number ๐›ผ, provided that the right-hand side is pointwisely defined on (0,+โˆž).

Remark 2.9. If ๐‘ฅ,๐‘ฆโˆถ(0,+โˆž)โ†’โ„ with order ๐›ผ>0, then ๐’Ÿ๐›ผ(๐‘ฅ(๐‘ก)+๐‘ฆ(๐‘ก))=๐’Ÿ๐›ผ๐‘ฅ(๐‘ก)+๐’Ÿ๐›ผ๐‘ฆ(๐‘ก).(2.8)

Lemma 2.10 (see [27]). (1) If ๐‘ฅโˆˆ๐ฟ1(0,1),๐œŒ>๐œŽ>0, then ๐ผ๐œŒ๐ผ๐œŽ๐‘ฅ(๐‘ก)=๐ผ๐œŒ+๐œŽ๐‘ฅ(๐‘ก),๐’Ÿ๐œŽ๐ผ๐œŒ๐‘ฅ(๐‘ก)=๐ผ๐œŒโˆ’๐œŽ๐‘ฅ(๐‘ก),๐’Ÿ๐œŽ๐ผ๐œŽ๐‘ฅ(๐‘ก)=๐‘ฅ(๐‘ก).(2.9)
(2) If ๐œŒ>0,๐œŽ>0, then ๐’Ÿ๐œŒ๐‘ก๐œŽโˆ’1=ฮ“(๐œŽ)๐‘กฮ“(๐œŽโˆ’๐œŒ)๐œŽโˆ’๐œŒโˆ’1.(2.10)

Lemma 2.11 (see [27]). Assume that ๐‘ฅโˆˆ๐ถ(0,1)โˆฉ๐ฟ1(0,1) with a fractional derivative of order ๐›ผ>0. Then ๐ผ๐›ผ๐’Ÿ๐›ผ๐‘ฅ(๐‘ก)=๐‘ฅ(๐‘ก)+๐‘1๐‘ก๐›ผโˆ’1+๐‘2๐‘ก๐›ผโˆ’2+โ‹ฏ+๐‘๐‘›๐‘ก๐›ผโˆ’๐‘›,(2.11) where ๐‘๐‘–โˆˆโ„(๐‘–=1,2,โ€ฆ,๐‘›), ๐‘› is the smallest integer greater than or equal to ๐›ผ.

Noticing that 2<๐›ผโˆ’๐œ‡๐‘›โˆ’1โ‰ค๐‘›โˆ’๐œ‡๐‘›โˆ’1<3, let โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘˜(๐‘ก,๐‘ )=(๐‘ก(1โˆ’๐‘ ))๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1โˆ’(๐‘กโˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ,0โ‰ค๐‘ โ‰ค๐‘กโ‰ค1,(๐‘ก(1โˆ’๐‘ ))๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ,0โ‰ค๐‘กโ‰ค๐‘ โ‰ค1,(2.12) by [28], for ๐‘ก,๐‘ โˆˆ[0,1], one has ๐‘ก๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1(1โˆ’๐‘ก)๐‘ (1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธโ‰ค๐‘˜(๐‘ก,๐‘ )โ‰ค๐‘ (1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ.(2.13)

Lemma 2.12. If 2<๐›ผโˆ’๐œ‡๐‘›โˆ’1<3 and ๐œŒโˆˆ๐ฟ1[0,1], then the boundary-value problem ๐’Ÿ๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘ค(๐‘ก)+๐œŒ(๐‘ก)=0,๐‘ค(0)=๐‘ค๎…ž(0)=0,๐‘ค(1)=๐‘โˆ’2๎“๐‘—=1๐‘Ž๐‘—๐‘ค๎€ท๐œ‰๐‘—๎€ธ,(2.14) has the unique solution ๎€œ๐‘ค(๐‘ก)=10๐พ(๐‘ก,๐‘ )๐œŒ(๐‘ )๐‘‘๐‘ ,(2.15) where ๐‘ก๐พ(๐‘ก,๐‘ )=๐‘˜(๐‘ก,๐‘ )+๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1โˆ‘1โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1๐‘โˆ’2๎“๐‘—=1๐‘Ž๐‘—๐‘˜๎€ท๐œ‰๐‘—๎€ธ,,๐‘ (2.16) is the Green function of the boundary-value problem (2.14).

Proof. By applying Lemma 2.11, we may reduce (2.14) to an equivalent integral equation: ๐‘ค(๐‘ก)=โˆ’๐ผ๐›ผโˆ’๐œ‡๐‘›โˆ’1๐œŒ(๐‘ก)+๐‘1๐‘ก๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1+๐‘2๐‘ก๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’2+๐‘3๐‘ก๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’3,๐‘1,๐‘2,๐‘3โˆˆโ„.(2.17) Note that ๐‘ค(0)=๐‘คโ€ฒ(0)=0 and (2.17), we have ๐‘2=๐‘3=0. Consequently the general solution of (2.14) is ๐‘ค(๐‘ก)=โˆ’๐ผ๐›ผโˆ’๐œ‡๐‘›โˆ’1๐œŒ(๐‘ก)+๐‘1๐‘ก๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1.(2.18) By (2.18) and Lemma 2.10, we have ๐‘ค(๐‘ก)=โˆ’๐ผ๐›ผโˆ’๐œ‡๐‘›โˆ’1๐œŒ(๐‘ก)+๐‘1๐‘ก๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1๎€œ=โˆ’๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ๐œŒ(๐‘ )๐‘‘๐‘ +๐‘1๐‘ก๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1.(2.19) So, ๎€œ๐‘ค(1)=โˆ’10(1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ๐œŒ(๐‘ )๐‘‘๐‘ +๐‘1,(2.20) and for ๐‘—=1,2,โ€ฆ,๐‘โˆ’2, ๐‘ค๎€ท๐œ‰๐‘—๎€ธ๎€œ=โˆ’๐œ‰๐‘—0๎€ท๐œ‰๐‘—๎€ธโˆ’๐‘ ๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ๐œŒ(๐‘ )๐‘‘๐‘ +๐‘1๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1.(2.21) By โˆ‘๐‘ค(1)=๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐‘ค(๐œ‰๐‘—), combining with (2.20) and (2.21), we obtain ๐‘1=โˆซ10(1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1โˆ‘๐œŒ(๐‘ )๐‘‘๐‘ โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—โˆซ๐œ‰๐‘—0๎€ท๐œ‰๐‘—๎€ธโˆ’๐‘ ๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1๐œŒ(๐‘ )๐‘‘๐‘ ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ๎‚€โˆ‘1โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1๎‚.(2.22) So, the unique solution of problem (2.14) is ๎€œ๐‘ค(๐‘ก)=โˆ’๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ+๐‘ก๐œŒ(๐‘ )๐‘‘๐‘ ๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1โˆ‘1โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1๎ƒฏ๎€œ10(1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ๐œŒ(๐‘ )๐‘‘๐‘ โˆ’๐‘โˆ’2๎“๐‘—=1๐‘Ž๐‘—๎€œ๐œ‰๐‘—0๎€ท๐œ‰๐‘—๎€ธโˆ’๐‘ ๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ๎ƒฐ๎€œ๐œŒ(๐‘ )๐‘‘๐‘ =โˆ’๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ๎€œ๐œŒ(๐‘ )๐‘‘๐‘ +10(1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1๐‘ก๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ+๐‘ก๐œŒ(๐‘ )๐‘‘๐‘ ๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1โˆ‘1โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1๐‘โˆ’2๎“๐‘—=1๐‘Ž๐‘—๎€œ10(1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธโˆ’๐‘ก๐œŒ(๐‘ )๐‘‘๐‘ ๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1โˆ‘1โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1๐‘โˆ’2๎“๐‘—=1๐‘Ž๐‘—๎€œ๐œ‰๐‘—0๎€ท๐œ‰๐‘—๎€ธโˆ’๐‘ ๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ=๎€œ๐œŒ(๐‘ )๐‘‘๐‘ 10โŽ›โŽœโŽœโŽ๐‘ก๐‘˜(๐‘ก,๐‘ )+๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1โˆ‘1โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1๐‘โˆ’2๎“๐‘—=1๐‘Ž๐‘—๐‘˜๎€ท๐œ‰๐‘—๎€ธโŽžโŽŸโŽŸโŽ =๎€œ,๐‘ ๐œŒ(๐‘ )๐‘‘๐‘ 10๐พ(๐‘ก,๐‘ )๐œŒ(๐‘ )๐‘‘๐‘ .(2.23) The proof is completed.

Lemma 2.13. The function ๐พ(๐‘ก,๐‘ ) has the following properties:
(1) ๐พ(๐‘ก,๐‘ )>0,๐‘“๐‘œ๐‘Ÿ๐‘ก,๐‘ โˆˆ(0,1);
(2) โ€‰๐พ(๐‘ก,๐‘ )โ‰ค๐‘€๐‘ (1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1,๐‘“๐‘œ๐‘Ÿ๐‘ก,๐‘ โˆˆ[0,1], where 1๐‘€=ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธโŽ›โŽœโŽœโŽโˆ‘1+๐‘โˆ’2๐‘—=1๐‘Ž๐‘—โˆ‘1โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1โŽžโŽŸโŽŸโŽ .(2.24)

Proof. It is obvious that (1) holds. In the following, we will prove (2). In fact, by (2.13), we have ๐‘ก๐พ(๐‘ก,๐‘ )=๐‘˜(๐‘ก,๐‘ )+๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1โˆ‘1โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1๐‘โˆ’2๎“๐‘—=1๐‘Ž๐‘—๐‘˜๎€ท๐œ‰๐‘—๎€ธโ‰ค๐‘ ,๐‘ (1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ+โˆ‘๐‘โˆ’2๐‘—=1๐‘Ž๐‘—โˆ‘1โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1๐‘ (1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธโ‰คโŽ›โŽœโŽœโŽโˆ‘1+๐‘โˆ’2๐‘—=1๐‘Ž๐‘—โˆ‘1โˆ’๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐œ‰๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘—โˆ’1โŽžโŽŸโŽŸโŽ ๐‘ (1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1ฮ“๎€ท๐›ผโˆ’๐œ‡๐‘›โˆ’1๎€ธ.(2.25) This completes the proof.

Now let us consider the following modified problems of the BVP (1.1): โˆ’๐’Ÿ๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก)=๐‘(๐‘ก)๐‘“(๐‘ก,๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐‘ข(๐‘ก),๐‘ข(๐‘ก)),๐‘ข(0)=0,๐‘ข๎…ž(0)=0,๐’Ÿ๐œ‡โˆ’๐œ‡๐‘›โˆ’1๐‘ข(1)=๐‘โˆ’2๎“๐‘—=1๐‘Ž๐‘—๐’Ÿ๐œ‡โˆ’๐œ‡๐‘›โˆ’1๐‘ข๎€ท๐œ‰๐‘—๎€ธ.(2.26)

Lemma 2.14. Let ๐‘ฅ(๐‘ก)=๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐‘ข(๐‘ก)โˆˆ๐ถ[0,1], Then (1.1) can be transformed into (2.26). Moreover, if ๐‘ขโˆˆ๐ถ([0,1],[0,+โˆž) is a solution of problem (2.26), then, the function ๐‘ฅ(๐‘ก)=๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก) is a positive solution of problem (1.1).

Proof. Substituting ๐‘ฅ(๐‘ก)=๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก) into (1.1), by the definition of the Riemann-Liouville fractional derivative and Lemmas 2.10 and 2.11, we obtain that ๐’Ÿ๐›ผ๐‘‘๐‘ฅ(๐‘ก)=๐‘›๐‘‘๐‘ก๐‘›๐ผ๐‘›โˆ’๐›ผ๐‘‘๐‘ฅ(๐‘ก)=๐‘›๐‘‘๐‘ก๐‘›๐ผ๐‘›โˆ’๐›ผ๐ผ๐œ‡๐‘›โˆ’1=๐‘‘๐‘ข(๐‘ก)๐‘›๐‘‘๐‘ก๐‘›๐ผ๐‘›โˆ’๐›ผ+๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก)=๐’Ÿ๐›ผโˆ’๐œ‡๐‘›โˆ’1๐’Ÿ๐‘ข(๐‘ก),๐œ‡1๐‘ฅ(๐‘ก)=๐’Ÿ๐œ‡1๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก)=๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐’Ÿ๐‘ข(๐‘ก),๐œ‡2๐‘ฅ(๐‘ก)=๐’Ÿ๐œ‡2๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก)=๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡2๐’Ÿ๐‘ข(๐‘ก),โ‹ฎโ‹ฎ๐œ‡๐‘›โˆ’2๐‘ฅ(๐‘ก)=๐’Ÿ๐œ‡๐‘›โˆ’2๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก)=๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐’Ÿ๐‘ข(๐‘ก),๐œ‡๐‘›โˆ’1๐‘ฅ(๐‘ก)=๐’Ÿ๐œ‡๐‘›โˆ’1๐ผ๐œ‡๐‘›โˆ’1๐’Ÿ๐‘ข(๐‘ก)=๐‘ข(๐‘ก),๐œ‡๐‘›โˆ’1+1๐‘ฅ๎€บ๐’Ÿ(๐‘ก)=๐œ‡๐‘›โˆ’1๐ผ๐œ‡๐‘›โˆ’1๐‘ข๎€ป(๐‘ก)๎…ž=๐‘ข๎…ž(๐‘ก).(2.27) Also, we have ๐’Ÿ๐œ‡๐‘›โˆ’1๐‘ฅ(0)=๐‘ข(0)=0,๐’Ÿ๐œ‡๐‘›โˆ’1+1๐‘ฅ(0)=๐‘ขโ€ฒ(0)=0, and it follows from ๐’Ÿ๐œ‡๐‘›โˆ’1๐‘ฅ(๐‘ก)=๐‘ข(๐‘ก) that โˆ‘๐‘ข(1)=๐‘โˆ’2๐‘—=1๐‘Ž๐‘—๐‘ข(๐œ‰๐‘—). Hence, by ๐‘ฅ(๐‘ก)=๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐‘ขโˆˆ๐ถ[0,1], (1.1) is transformed into (2.14).
Now, let ๐‘ขโˆˆ๐ถ([0,1,0,+โˆž)) be a solution of problem (2.26). Then, by Lemma 2.10, (2.26), and (2.27), one has โˆ’๐’Ÿ๐›ผ๐‘‘๐‘ฅ(๐‘ก)=โˆ’๐‘›๐‘‘๐‘ก๐‘›๐ผ๐‘›โˆ’๐›ผ๐‘‘๐‘ฅ(๐‘ก)=โˆ’๐‘›๐‘‘๐‘ก๐‘›๐ผ๐‘›โˆ’๐›ผ๐ผ๐œ‡๐‘›โˆ’1๐‘‘๐‘ข(๐‘ก)=โˆ’๐‘›๐‘‘๐‘ก๐‘›๐ผ๐‘›โˆ’๐›ผ+๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก)=โˆ’๐’Ÿ๐›ผโˆ’๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก)=๐‘(๐‘ก)๐‘“(๐‘ก,๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐‘ข(๐‘ก),๐‘ข(๐‘ก))=๐‘(๐‘ก)๐‘“(๐‘ก,๐‘ฅ(๐‘ก),๐’Ÿ๐œ‡1๐‘ฅ(๐‘ก),๐’Ÿ๐œ‡2๐‘ฅ(๐‘ก),โ€ฆ,๐’Ÿ๐œ‡๐‘›โˆ’1๐‘ฅ(๐‘ก)),0<๐‘ก<1.(2.28) Noticing that ๐ผ๐›ผ1๐‘ข(๐‘ก)=๎€œฮ“(๐›ผ)๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1๐‘ข(๐‘ )๐‘‘๐‘ ,(2.29) which implies that ๐ผ๐›ผ๐‘ข(0)=0, from (2.27), for ๐‘–=1,2,โ€ฆ,๐‘›โˆ’1, we have ๐’Ÿ๐œ‡๐‘–๐‘ฅ(0)=0,๐’Ÿ๐œ‡๐‘›โˆ’1+1๐‘ฅ(0)=0,๐’Ÿ๐œ‡๐‘›โˆ’1๐‘ฅ(1)=๐‘โˆ’2๎“๐‘—=1๐‘Ž๐‘—๐’Ÿ๐œ‡๐‘›โˆ’1๐‘ฅ๎€ท๐œ‰๐‘—๎€ธ.(2.30)
Moreover, it follows from the monotonicity and property of ๐ผ๐œ‡๐‘›โˆ’1 that ๐ผ๐œ‡๐‘›โˆ’1[],[๐‘ขโˆˆ๐ถ(0,10,+โˆž)).(2.31)
Consequently, ๐‘ฅ(๐‘ก)=๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก) is a positive solution of problem (1.1).

In the following let us list some assumptions to be used in the rest of this paper.

(H1)โ€‰โ€‰๐‘โˆถ(0,1)โ†’[0,+โˆž) is continuous, ๐‘(๐‘ก)โ‰ข0 on any subinterval of (0,1), and ๎€œ10๐‘(๐‘ )๐‘‘๐‘ <+โˆž.(2.32)

(H2)โ€‰โ€‰๐‘“โˆถ[0,1]ร—โ„๐‘›โ†’(โˆ’โˆž,+โˆž) is continuous.

In order to use Lemma 2.5, let ๐ธ=๐ถ[0,1] be our real Banach space with the norm โ€–๐‘ขโ€–=max๐‘กโˆˆ[0,1]|๐‘ข(๐‘ก)| and ๐‘ƒ={๐‘ขโˆˆ๐ถ[0,1]โˆถ๐‘ข(๐‘ก)โ‰ฅ0,forall๐‘กโˆˆ[0,1]}, then ๐‘ƒ is a total cone in ๐ธ.

Define two linear operators ๐ฟ,๐ฝโˆถ๐ถ[0,1]โ†’๐ถ[0,1] by ๎€œ(๐ฟ๐‘ข)(๐‘ก)=10[][],๎€œ๐พ(๐‘ก,๐‘ )๐‘(๐‘ )๐‘ข(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ0,1,๐‘ขโˆˆ๐ถ0,1(2.33)(๐ฝ๐‘ข)(๐‘ก)=10๐พ[][],(๐‘ ,๐‘ก)๐‘(๐‘ )๐‘ข(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ0,1,๐‘ขโˆˆ๐ถ0,1(2.34) and define a nonlinear operator ๐ดโˆถ๐ถ[0,1]โ†’๐ถ[0,1] by ๎€œ(๐ด๐‘ข)(๐‘ก)=10๐พ(๐‘ก,๐‘ )๐‘(๐‘ )๐‘“(๐‘ก,๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ ),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ ),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2[].๐‘ข(๐‘ ),๐‘ข(๐‘ ))๐‘‘๐‘ ,๐‘กโˆˆ0,1(2.35)

Lemma 2.15. Assume (H1) holds. Then(i)๐ฟ,๐ฝโˆถ๐ถ[0,1]โ†’๐ถ[0,1] are completely continuous positive linear operators with the first eigenvalue ๐‘Ÿ>0 and ๐œ†>0, respectively.(ii)๐ฟ satisfies ๐ฟ(๐‘ƒ)โŠ‚๐‘ƒ(๐‘”1,๐›ฟ).

Proof. (i) By using the similar method of paper [22], it is easy to know that ๐ฟ,๐ฝโˆถ๐ถ[0,1]โ†’๐ถ[0,1] are completely continuous positive linear operators. In the following, by using the Krein-Rutmannโ€™s theorem, we prove that ๐ฟ,๐ฝ have the first eigenvalue ๐‘Ÿ>0 and ๐œ†>0, respectively.
In fact, it is obvious that there is ๐‘ก1โˆˆ(0,1) such that ๐พ(๐‘ก1,๐‘ก1)๐‘(๐‘ก1)>0. Thus there exists [๐‘Ž,๐‘]โŠ‚(0,1) such that ๐‘ก1โˆˆ(๐‘Ž,๐‘) and ๐พ(๐‘ก,๐‘ )๐‘(๐‘ )>0 for all ๐‘ก,๐‘ โˆˆ[๐‘Ž,๐‘]. Choose ๐œ“โˆˆ๐‘ƒ such that ๐œ“(๐‘ก1)>0 and ๐œ“(๐‘ก)=0 for all ๐‘กโˆ‰[๐‘Ž,๐‘]. Then for ๐‘กโˆˆ[๐‘Ž,๐‘], ๎€œ(๐ฟ๐œ“)(๐‘ก)=10๎€œ๐พ(๐‘ก,๐‘ )๐‘(๐‘ )๐œ“(๐‘ )๐‘‘๐‘ โ‰ฅ๐‘๐‘Ž๐พ(๐‘ก,๐‘ )๐‘(๐‘ )๐œ“(๐‘ )๐‘‘๐‘ >0.(2.36)
So there exists ๐œˆ>0 such that ๐œˆ(๐ฟ๐œ“)(๐‘ก)โ‰ฅ๐œ“(๐‘ก) for ๐‘กโˆˆ[0,1]. It follows from Lemma 2.3 that the spectral radius ๐‘Ÿ1โ‰ 0. Thus corresponding to ๐‘Ÿ=๐‘Ÿ1โˆ’1, the first eigenvalue of ๐ฟ, and ๐ฟ has a positive eigenvector ๐œ‘1 such that ๐‘Ÿ๐ฟ๐œ‘1=๐œ‘1.(2.37)
In the same way, ๐ฝ has a positive first eigenvalue ๐œ† and a positive eigenvector ๐œ‘2 corresponding to the first eigenvalue ๐œ†, which satisfy ๐œ†๐ฝ๐œ‘2=๐œ‘2.(2.38)
(ii) Notice that ๐พ(๐‘ก,0)=๐พ(๐‘ก,1)โ‰ก0 for ๐‘กโˆˆ[0,1], by ๐œ†๐ฝ๐œ‘2=๐œ‘2 and (2.12)โ€“(2.16), we have ๐œ‘2(0)=๐œ‘2(1)=0. This implies that ๐œ‘๎…ž2(0)>0 and ๐œ‘๎…ž2(1)<0 (see [29]). Define a function ๐œ’ on [0,1] by โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐œ‘๐œ’(๐‘ )=๎…ž2๐œ‘(0),๐‘ =0,2(๐‘ )(1โˆ’๐‘ )๐‘ ,0<๐‘ <1,โˆ’๐œ‘๎…ž2(1),๐‘ =1.(2.39)
Then ๐œ’ is continuous on [0,1] and ๐œ’(๐‘ )>0 for all ๐‘ โˆˆ[0,1]. So, there exist ๐›ฟ1,๐›ฟ2>0 such that ๐›ฟ1โ‰ค๐œ’(๐‘ )โ‰ค๐›ฟ2 for all ๐‘ โˆˆ[0,1]. Thus ๐›ฟ1(1โˆ’๐‘ )๐‘ โ‰ค๐œ‘2(๐‘ )โ‰ค๐›ฟ2(1โˆ’๐‘ )๐‘ ,(2.40) for all ๐‘ โˆˆ[0,1].
Let ๐ฟโˆ— be the dual operator of ๐ฟ, we will show that there exists ๐‘”1โˆˆ๐‘ƒโˆ—โงต{๐œƒ} such that ๐œ†๐ฟโˆ—๐‘”1=๐‘”1.(2.41) In fact, let ๐‘”1๎€œ(๐‘ข)=10๐‘(๐‘ก)๐œ‘2(๐‘ก)๐‘ข(๐‘ก)๐‘‘๐‘กโˆ€๐‘ขโˆˆ๐ธ.(2.42) Then by (H1) and (2.40), we have ๎€œ10๐‘(๐‘ก)๐œ‘2(๐‘ก)๐‘ข(๐‘ก)๐‘‘๐‘กโ‰ค๐›ฟ2๎€œโ€–๐‘ขโ€–10๐‘ก(1โˆ’๐‘ก)๐‘(๐‘ก)๐‘‘๐‘กโ‰ค๐›ฟ2๎€œโ€–๐‘ขโ€–10๐‘(๐‘ก)๐‘‘๐‘ก<+โˆž,(2.43) which implies that ๐‘”1 is well defined. We state that ๐‘”1 of (2.42) satisfies (2.41). In fact, by (2.40), (2.41), and interchanging the order of integration, for any ๐‘ ,๐‘กโˆˆ[0,1], we have ๐œ†โˆ’1๐‘”1๎€œ(๐‘ข)=10๎€ท๐œ†๐‘(๐‘ก)โˆ’1๐œ‘2๎€ธ๎€œ(๐‘ก)๐‘ข(๐‘ก)๐‘‘๐‘ก=10๎€ท๐‘(๐‘ก)๐ฝ๐œ‘2๎€ธ=๎€œ(๐‘ก)๐‘ข(๐‘ก)๐‘‘๐‘ก10๐‘๎€œ(๐‘ก)๐‘ข(๐‘ก)10๐พ(๐‘ ,๐‘ก)๐‘(๐‘ )๐œ‘2=๎€œ(๐‘ )๐‘‘๐‘ ๐‘‘๐‘ก10๐‘(๐‘ )๐œ‘2๎€œ(๐‘ )10๐พ=๎€œ(๐‘ ,๐‘ก)๐‘(๐‘ก)๐‘ข(๐‘ก)๐‘‘๐‘ก๐‘‘๐‘ 10๐‘(๐‘ )๐œ‘2(๐‘ )(๐ฟ๐‘ข)(๐‘ )๐‘‘๐‘ =๐‘”1๎€ท๐ฟ(๐ฟ๐‘ข)=โˆ—๐‘”1๎€ธ(๐‘ข)โˆ€๐‘ขโˆˆ๐ธ.(2.44) So (2.41) holds.
In the following we prove that ๐ฟ(๐‘ƒ)โŠ‚๐‘ƒ(๐‘”1,๐›ฟ). In fact, by (2.41) and ๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1>1, we have ๐œ‘2(๐‘ )โ‰ฅ๐›ฟ1(1โˆ’๐‘ )๐‘ โ‰ฅ๐›ฟ1(1โˆ’๐‘ )๐›ผโˆ’๐œ‡๐‘›โˆ’1โˆ’1๐‘ โ‰ฅ๐›ฟ1๐‘€โˆ’1[].๐พ(๐‘ก,๐‘ ),๐‘ก,๐‘ โˆˆ0,1(2.45) Take ๐›ฟ=๐›ฟ1๐‘€โˆ’1๐œ†โˆ’1>0 in (2.2). For any ๐‘ขโˆˆ๐‘ƒ, by (2.44), (2.45), we have ๐‘”1(๐ฟ๐‘ข)=๐œ†โˆ’1๐‘”1(๐‘ข)=๐œ†โˆ’1๎€œ10๐‘(๐‘ )๐œ‘2(๐‘ )๐‘ข(๐‘ )๐‘‘๐‘ โ‰ฅ๐›ฟ1๐‘€โˆ’1๐œ†โˆ’1๎€œ10๐พ[].(๐‘ก,๐‘ )๐‘(๐‘ )๐‘ข(๐‘ )๐‘‘๐‘ =๐›ฟ(๐ฟ๐‘ข)(๐‘ก)โˆ€๐‘กโˆˆ0,1(2.46) Hence, ๐‘”1(๐ฟ๐‘ข)โ‰ฅ๐›ฟ||๐ฟ๐‘ข||, that is, ๐ฟ(๐‘ƒ)โŠ‚๐‘ƒ(๐‘”1,๐›ฟ). The proof is completed.

3. Main Result

Theorem 3.1. Assume that (H1)(H2) hold, and the following conditions are satisfied.
(H3) There exist nonnegative continuous functions ๐‘,๐‘โˆถ[0,1]โ†’(0,+โˆž) and a nondecreasing continuous function โ„Žโˆถโ„๐‘›โ†’[0,+โˆž) satisfying limโˆ‘๐‘›๐‘–=1||๐‘ฅ๐‘–||โ†’+โˆžโ„Ž๎€ท๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธโˆ‘๐‘›๐‘–=1||๐‘ฅ๐‘–||๐‘“๎€ท=0,๐‘ก,๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโ‰ฅโˆ’๐‘(๐‘ก)โˆ’๐‘(๐‘ก)โ„Ž1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ,โˆ€๐‘ฅ๐‘–โˆˆโ„.(3.1)
(H4) ๐‘“ also satisfies liminfโˆ‘๐‘›๐‘ฅ๐‘–๐‘–=1โ†’+โˆžโˆ‘๐‘›โˆ’1๐‘–=1๐‘ฅ๐‘–โ‰ฅ0๐‘“๎€ท๐‘ก,๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธโˆ‘๐‘›๐‘–=1๐‘ฅ๐‘–>๐œ†limsupโˆ‘๐‘›๐‘–=1|๐‘ฅ๐‘–|โ†’0||๐‘“๎€ท๐‘ก,๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ||โˆ‘๐‘›๐‘–=1||๐‘ฅ๐‘–||<๐œ†,(3.2) uniformly on ๐‘กโˆˆ[0,1], where ๐œ† is the first eigenvalue of the operator ๐ฝ defined by (2.34).
Then the singular fractional-order boundary-value problem (1.1) has at least one nontrivial solution.

Proof. According to Lemma 2.15, ๐ฟ satisfies ๐ฟ(๐‘ƒ)โŠ‚๐‘ƒ(๐‘”1,๐›ฟ). Let (๐‘‡๐‘ข)(๐‘ก)=๐‘โ„Ž(๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐‘ข(๐‘ก),๐‘ข(๐‘ก))for๐‘ขโˆˆ๐ธ,(3.3) where ๐‘=max๐‘กโˆˆ[0,1]๐‘(๐‘ก). It follows from (H3) that ๐‘‡โˆถ๐ธโ†’๐‘ƒ is a continuous operator. Note that ๐ผ๐œ‡๐‘›โˆ’1๎€œ๐‘ข(๐‘ก)=๐‘ก0(๐‘กโˆ’๐‘ )๐œ‡๐‘›โˆ’1โˆ’1๐‘ข(๐‘ )ฮ“๎€ท๐œ‡๐‘›โˆ’1๎€ธ๐‘‘๐‘ โ‰คโ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1๎€ธ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘–๎€œ๐‘ข(๐‘ก)=๐‘ก0(๐‘กโˆ’๐‘ )๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘–โˆ’1๐‘ข(๐‘ )ฮ“๎€ท๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘–๎€ธ๐‘‘๐‘ โ‰คโ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘–๎€ธ,๐‘–=1,2,โ€ฆ,๐‘›โˆ’2.(3.4) Thus from the monotone assumption of โ„Ž on ๐‘ฅ๐‘–, we have โ„Ž(๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๎ƒฉ๐‘ข(๐‘ก),๐‘ข(๐‘ก))โ‰คโ„Žโ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1๎€ธ,โ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๎€ธ,โ€ฆ,โ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๎€ธ๎ƒช[],,โ€–๐‘ขโ€–,forany๐‘ขโˆˆ๐ธ,๐‘กโˆˆ0,1(3.5) which implies that โ€–๐‘‡๐‘ขโ€–=max[]๐‘กโˆˆ0,1{๐‘โ„Ž(๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๎ƒฉ๐‘ข(๐‘ก),๐‘ข(๐‘ก))}โ‰ค๐‘โ„Žโ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1๎€ธ,โ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๎€ธ,โ€ฆ,โ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๎€ธ๎ƒช,โ€–๐‘ขโ€–forany๐‘ขโˆˆ๐ธ.(3.6) Let 1๐œ=ฮ“๎€ท๐œ‡๐‘›โˆ’1๎€ธ+1ฮ“๎€ท๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๎€ธ1+โ‹ฏ+ฮ“๎€ท๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๎€ธ+1,(3.7) then limโ€–๐‘ขโ€–โ†’+โˆžโ€–๐‘‡๐‘ขโ€–โ€–๐‘ขโ€–โ‰คlimโ€–๐‘ขโ€–โ†’+โˆž๎€ท๎€ท๐œ‡๐‘โ„Žโ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1๎€ธ๎€ท๐œ‡,โ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1โˆ’๐œ‡1๎€ธ๎€ท๐œ‡,โ€ฆ,โ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๎€ธ๎€ธ,โ€–๐‘ขโ€–โ€–๐‘ขโ€–=limโ€–๐‘ขโ€–โ†’+โˆž๎€ท๎€ท๐œ‡๐‘๐œโ„Žโ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1๎€ธ๎€ท๐œ‡,โ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1โˆ’๐œ‡1๎€ธ๎€ท๐œ‡,โ€ฆ,โ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๎€ธ๎€ธ,โ€–๐‘ขโ€–๐œโ€–๐‘ขโ€–โ‰คlimโ€–๐‘ขโ€–โ†’+โˆž๎€ท๎€ท๐œ‡๐‘๐œโ„Žโ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1๎€ธ๎€ท๐œ‡,โ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1โˆ’๐œ‡1๎€ธ๎€ท๐œ‡,โ€ฆ,โ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๎€ธ๎€ธ,โ€–๐‘ขโ€–๎€ท๐œ‡โ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1๎€ธ๎€ท๐œ‡+โ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1โˆ’๐œ‡1๎€ธ๎€ท๐œ‡+โ‹ฏ+โ€–๐‘ขโ€–/ฮ“๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๎€ธ+โ€–๐‘ขโ€–=๐‘๐œlimโˆ‘๐‘›๐‘–=1||๐‘ฅ๐‘–||โ†’+โˆžโ„Ž๎€ท๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธโˆ‘๐‘›๐‘–=1||๐‘ฅ๐‘–||=0,(3.8) that is, limโ€–๐‘ขโ€–โ†’+โˆžโ€–๐‘‡๐‘ขโ€–โ€–๐‘ขโ€–=0.(3.9) Hence ๐‘‡ satisfies condition (A1) in Lemma 2.5.
Next take ๐‘ข0(๐‘ก)โ‰ก๐‘(๐‘ก), and (๐น๐‘ข)(๐‘ก)=๐‘“(๐‘ก,๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐‘ข(๐‘ก),๐‘ข(๐‘ก)) for ๐‘ขโˆˆ๐ธ. Then it follows from (H3) that ๐น๐‘ข+๐‘ข0+๐‘‡๐‘ข=๐‘“(๐‘ก,๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐‘ข(๐‘ก),๐‘ข(๐‘ก))+๐‘(๐‘ก)+๐‘โ„Ž(๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐‘ข(๐‘ก),๐‘ข(๐‘ก))โ‰ฅ0,(3.10) that yields ๐น๐‘ข+๐‘ข0+๐‘‡๐‘ขโˆˆ๐‘ƒ,โˆ€๐‘ขโˆˆ๐ธ,(3.11) namely, condition (A2) in Lemma 2.5 holds.
From (H4), there exists ๐œ€>0 and a sufficiently large ๐‘™1>0 such that, for any ๐‘กโˆˆ[0,1], ๐‘“๎€ท๐‘ก,๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโ‰ฅ๐œ†(1+๐œ€)1+๐‘ฅ2+โ‹ฏ+๐‘ฅ๐‘›๎€ธโ‰ฅ๐œ†(1+๐œ€)๐‘ฅ๐‘›,๐‘ฅ1+๐‘ฅ2+โ‹ฏ+๐‘ฅ๐‘›>๐‘™1.(3.12) Combining (H3) with (3.12), there exists ๐‘1โ‰ฅ0 such that ๐น๐‘ข=๐‘“(๐‘ก,๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐‘ข(๐‘ก),๐‘ข(๐‘ก))โ‰ฅ๐œ†(1+๐œ€)๐‘ข(๐‘ก)โˆ’๐‘1โˆ’๐‘โ„Ž(๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐‘ข(๐‘ก),๐‘ข(๐‘ก)),โˆ€๐‘ขโˆˆ๐ธ,(3.13) that is, F๐‘ขโ‰ฅ๐œ†(1+๐œ€)๐‘ขโˆ’๐‘1โˆ’๐‘‡๐‘ขโˆ€๐‘ขโˆˆ๐ธ.(3.14) As ๐ฟ is a positive linear operator, it follows from (3.14) that (๐ฟ๐น๐‘ข)(๐‘ก)โ‰ฅ๐œ†(1+๐œ€)(๐ฟ๐‘ข)(๐‘ก)โˆ’๐ฟ๐‘1[].โˆ’(๐ฟ๐‘‡๐‘ข)(๐‘ก),โˆ€๐‘กโˆˆ0,1(3.15) So condition (A3) in Lemma 2.5 holds. According to Lemma 2.5, there exists a sufficiently large number ๐‘…>0 such that ๎€ทdeg๐ผโˆ’๐ด,๐ต๐‘…๎€ธ,๐œƒ=0.(3.16)
On the other hand, it follows from (H4) that there exist 0<๐œ€<1 and 0<๐‘Ÿ<๐‘…, for any ๐‘กโˆˆ[0,1], such that ||๐‘“๎€ท๐‘ก,๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ||โ‰ค1โˆ’๐œ€โˆซ๐‘€๐œ10๎€ท||๐‘ฅ๐‘(๐‘ )๐‘‘๐‘ 1||+||๐‘ฅ2||||๐‘ฅ+โ‹ฏ+๐‘›||๎€ธ,||๐‘ฅ1||+||๐‘ฅ2||||๐‘ฅ+โ‹ฏ+๐‘›||<๐‘Ÿ.(3.17) Thus for any ๐‘ขโˆˆ๐ธ with ||๐‘ข||โ‰ค๐‘Ÿ/๐œโ‰ค๐‘Ÿโ‰ค๐‘…, we have ||๐ผ๐œ‡๐‘›โˆ’1||+||๐ผ๐‘ข(๐‘ก)๐œ‡๐‘›โˆ’1โˆ’๐œ‡1||||๐ผ๐‘ข(๐‘ก)+โ‹ฏ+๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2||+||||โ‰ค๐‘ข(๐‘ก)๐‘ข(๐‘ก)โ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1๎€ธ+โ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๎€ธ+โ‹ฏ+โ€–๐‘ขโ€–ฮ“๎€ท๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๎€ธ+โ€–๐‘ขโ€–โ‰ค๐œโ€–๐‘ขโ€–โ‰ค๐‘Ÿ.(3.18) By (3.17), for any ๐‘กโˆˆ[0,1], we have ||๐‘“(๐‘ก,๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก),๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡1๐‘ข(๐‘ก),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2||โ‰ค๐‘ข(๐‘ก),๐‘ข(๐‘ก))1โˆ’๐œ€โˆซ๐‘€๐œ10๎€ท||๐ผ๐‘(๐‘ )๐‘‘๐‘ ๐œ‡๐‘›โˆ’1||+||๐ผ๐‘ข(๐‘ก)๐œ‡๐‘›โˆ’1โˆ’๐œ‡1||||๐ผ๐‘ข(๐‘ก)+โ‹ฏ+๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2||+||||๎€ธ.๐‘ข(๐‘ก)๐‘ข(๐‘ก)(3.19) Thus if there exist ๐‘ข1โˆˆ๐œ•๐ต๐‘Ÿ/๐œ and ๐œ‡1โˆˆ[0,1] such that ๐‘ข1=๐œ‡1๐ด๐‘ข1, then by (3.19), we have ๐‘”1๎€ทโ€–โ€–๐‘ข1โ€–โ€–๎€ธ=๐‘”1๎€ท๐œ‡1โ€–โ€–๐ด๐‘ข1โ€–โ€–๎€ธ=๐œ‡1๐‘”1๎€ทโ€–โ€–๐ด๐‘ข1โ€–โ€–๎€ธ=๐œ‡1๐‘”1๎ƒฉmax0โ‰ค๐‘กโ‰ค1||||๎€œ10๎€ท๐พ(๐‘ก,๐‘ )๐‘(๐‘ )๐‘“๐‘ ,๐ผ๐œ‡๐‘›โˆ’1๐‘ข1(๐‘ ),โ€ฆ,๐ผ๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐‘ข1(๐‘ ),๐‘ข1๎€ธ||||๎ƒชโ‰ค(๐‘ )๐‘‘๐‘ 1โˆ’๐œ€โˆซ๐‘€๐œ10๐‘”๐‘(๐‘ )๐‘‘๐‘ 1๎‚ต๎€œ10๎€ท||๐ผ๐‘€๐‘(๐‘ )๐œ‡๐‘›โˆ’1๐‘ข1||||๐ผ(๐‘ )+โ‹ฏ+๐œ‡๐‘›โˆ’1โˆ’๐œ‡๐‘›โˆ’2๐‘ข1||+||๐‘ข(๐‘ )1||๎€ธ๎‚ถโ‰ค(๐‘ )๐‘‘๐‘ 1โˆ’๐œ€โˆซ๐‘€๐œ10๎€œ๐‘(๐‘ )๐‘‘๐‘ ๐‘€๐œ10๐‘(๐‘ )๐‘‘๐‘ ๐‘”1๎€ทโ€–โ€–๐‘ข1โ€–โ€–๎€ธ=(1โˆ’๐œ€)๐‘”1๎€ทโ€–โ€–๐‘ข1โ€–โ€–๎€ธ.(3.20) Therefore, ๐‘”1(โ€–๐‘ข1โ€–)โ‰ค0.
But ๐œ‘2(๐‘ก)>0 for all ๐‘กโˆˆ(0,1) by the maximum principle, and ๐‘ข1(๐‘ก) attains zero on isolated points by the Sturm theorem. Hence, from (2.42), ๐‘”1๎€ทโ€–โ€–๐‘ข1โ€–โ€–๎€ธ=๎€œ10๐‘(๐‘ก)๐œ‘2โ€–โ€–๐‘ข(๐‘ก)1โ€–โ€–๐‘‘๐‘ก>0.(3.21) This is a contradiction. Thus ๐‘ขโ‰ ๐œ‡๐ด๐‘ข,โˆ€๐‘ขโˆˆ๐œ•๐ต๐‘Ÿ,[].๐œ‡โˆˆ0,1(3.22) It follows from the homotopy invariance of the Leray-Shauder degree that ๎€ทdeg๐ผโˆ’๐ด,๐ต๐‘Ÿ/๐œ๎€ธ,๐œƒ=1.(3.23) By (3.16), (3.23), and the additivity of Leray-Shauder degree, we obtain ๎€ทdeg๐ผโˆ’๐ด,๐ต๐‘…โงต๐ต๐‘Ÿ/๐œ๎€ธ๎€ท,๐œƒ=deg๐ผโˆ’๐ด,๐ต๐‘…๎€ธ๎€ท,๐œƒโˆ’deg๐ผโˆ’๐ด,๐ต๐‘Ÿ/๐œ๎€ธ,๐œƒ=โˆ’1.(3.24) As a result, ๐ด has at least one fixed point ๐‘ข on ๐ต๐‘…โงต๐ต๐‘Ÿ/๐œ, namely, the BVP (1.1) has at least one nontrivial solution ๐‘ฅ(๐‘ก)=๐ผ๐œ‡๐‘›โˆ’1๐‘ข(๐‘ก).

Corollary 3.2. Assume that (H1), (H2), and (H4) hold. If the following condition is satisfied.
(H*3) There exists a nonnegative continuous functions ๐‘โˆถ[0,1]โ†’(0,+โˆž) such that ๐‘“๎€ท๐‘ก,๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ[]โ‰ฅโˆ’๐‘(๐‘ก),๐‘“๐‘œ๐‘Ÿ๐‘Ž๐‘›๐‘ฆ๐‘กโˆˆ0,1,โˆ€๐‘ฅ๐‘–โˆˆโ„,(3.25) then the singular higher multipoint boundary-value problems (1.1) have at least one nontrivial solution.

Corollary 3.3. Assume that (H1), (H2), and (H4) hold. If the following condition is satisfied.
(H3**) There exist three constants ๐‘>0,๐‘>0, and ๐›ผ๐‘–โˆˆ(0,1) such that ๐‘“๎€ท๐‘ก,๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธโ‰ฅโˆ’๐‘โˆ’๐‘๐‘›๎“๐‘–=1||๐‘ฅ๐‘–||๐›ผ๐‘–[],๐‘“๐‘œ๐‘Ÿ๐‘Ž๐‘›๐‘ฆ๐‘กโˆˆ0,1,โˆ€๐‘ฅ๐‘–โˆˆโ„,(3.26) then the singular higher multipoint boundary-value problems (1.1) have at least one nontrivial solution.

Remark 3.4. Noticing that the Green function of the BVP (1.1) is not symmetrical, which implies that the existence of nontrivial solutions of the BVP (1.1) cannot be obtained by Theoremโ€‰โ€‰2.1 in [22] and Theoremโ€‰โ€‰1 in [20]. It is interesting that we construct a new linear operator ๐ฝ instead of ๐พ in paper [22] and use its first eigenvalue and its corresponding eigenfunction to seek a linear continuous functional ๐‘” of ๐‘ƒ. As a result, we overcome the difficulty caused by the nonsymmetry of the Green function. In [24], the nonlinearity does not contain derivatives and a stronger condition is required, that is, โ„Ž must be an even function; here we omit this stronger assumption.

Remark 3.5. The results of [20โ€“22] is a special case of the Corollary 3.2 and Corollary 3.3 when ๐›ผ1(๐‘–=1,2,โ€ฆ,๐‘›) are integer and the nonlinear term does not involve derivatives of unknown functions.

Acknowledgments

This project is supported financially by Scientific Research Project of Zhejiang Education Department (no. Y201016244), also by Scientific Research Project of Wenzhou (no. G20110004), Natural Science Foundation of Zhejiang Province (No. 2012C31025) and the National Natural Science Foundation of China (11071141, 11126231, 21207103), and the Natural Science Foundation of Shandong Province of China (ZR2010AM017).

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